# Feasible Optimal Solutions of Electromagnetic Cloaking Problems by Chaotic Accelerated Particle Swarm Optimization

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Chaotic Accelerated Particle Swarm Optimization

#### 2.1. Particle Swarm Optimization Fundamentals

- Proximity: The swarm has the ability to complete simple/elementary computations. We mostly refer to time and space computations, since they concern the swarm’s “natural” environment;
- Quality: The swarm considers quality factors as well (e.g., safety);
- Diverse response: The swarm does not narrow down its strategies regarding the environment to an extreme degree, causing limitations. It ensures the ability to follow alternatives;
- Stability: The swarm does not alter its mode of behavior for every single change observed in its environment;
- Adaptability: The swarm alters its mode of behavior in response to environmental changes when this decision is ensured to be a beneficial one (the swarms are designed to have means of knowing so).

- Candidate solutions are represented by the particles’ positions $\mathbf{x}\in {\mathbb{R}}^{n}$;
- The particles move in the search space possessing velocity $\mathbf{u}\in {\mathbb{R}}^{n}$;
- There is shared knowledge in the swarm regarding the top solution(s) discovered by the swarm. The best solution of the whole swarm per iteration is usually referred to as the global best ${\mathbf{g}}^{*}$. There can be multiples if the algorithm follows such a topology;
- The velocity update formula is certainly affected by the global best solution and by other parameters. For example, the particle’s individual best position so far, ${\mathbf{x}}^{*}$, which is known as the local best;
- Some randomness is necessary to establish better solutions. It is very common to insert some randomness factors in the velocity update formula.
- The position updates after the velocity, commonly with respect to laws of the Newtonian (classical) mechanics;
- The algorithm converges when the particles have agreed to an optimal position. Additionally, there is a stopping criterion when the maximum number of iterations is met.

#### 2.2. Accelerated Particle Swarm Optimization (APSO)

#### 2.3. The Chaotic Accelerated Particle Swarm Optimization (CAPSO) Algorithm

Algorithm 1: Chaotic APSO (CAPSO) |

## 3. Optimization Problem

#### 3.1. Exact Solution of the Scattering Problem

#### 3.2. Strategy

## 4. Numerical Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

PSO | Particle Swarm Optimization |

APSO | Accelerated Particle Swarm Optimization |

CAPSO | Chaotic Accelerated Particle Swarm Optimization |

PEC | Perfect Electric Conducting |

SCS | Scattering Cross Section |

## Appendix A

**Table A1.**Values of the optimization variables for optimized designs A and B of Figure 2.

A ($\mathit{N}=3$) | B ($\mathit{N}=3$) | A ($\mathit{N}=4$) | B ($\mathit{N}=4$) | |
---|---|---|---|---|

${k}_{0}{a}_{1}$ | 8.12 | 8.09 | 8.24 | 8.66 |

${k}_{0}{a}_{2}$ | 7.36 | 7.30 | 7.86 | 7.70 |

${k}_{0}{a}_{3}$ | 6.65 | 6.60 | 7.30 | 6.89 |

${k}_{0}{a}_{4}$ | – | – | 6.67 | 6.75 |

${\u03f5}_{1}$ | 0.40 | 0.40 | 0.60 | 0.50 |

${\u03f5}_{2}$ | 3.87 | 4.15 | 0.40 | 2.93 |

${\u03f5}_{3}$ | 0.40 | 0.40 | 5.00 | 0.50 |

${\u03f5}_{4}$ | – | – | 0.60 | 0.53 |

${\mu}_{1}$ | 1.92 | 1.88 | 2.59 | 1.62 |

${\mu}_{2}$ | 0.40 | 0.40 | 0.86 | 0.50 |

${\mu}_{3}$ | 1.73 | 1.79 | 0.40 | 2.62 |

${\mu}_{4}$ | – | – | 1.25 | 0.97 |

**Table A2.**Values of the optimization variables for optimized designs A and B of Figure 4.

A ($\mathit{N}=3$) | B ($\mathit{N}=3$) | A ($\mathit{N}=4$) | B ($\mathit{N}=4$) | |
---|---|---|---|---|

${k}_{0}{a}_{1}$ | 8.10 | 7.54 | 8.17 | 7.68 |

${k}_{0}{a}_{2}$ | 7.41 | 7.24 | 7.46 | 7.32 |

${k}_{0}{a}_{3}$ | 6.67 | 6.60 | 6.87 | 6.62 |

${k}_{0}{a}_{4}$ | – | – | 6.76 | 6.40 |

${\u03f5}_{1}$ | 0.40 | 1.00 | 0.40 | 0.68 |

${\u03f5}_{2}$ | 3.73 | 4.71 | 3.85 | 5.00 |

${\u03f5}_{3}$ | 0.40 | 0.40 | 1.35 | 0.40 |

${\u03f5}_{4}$ | – | – | 0.40 | 1.96 |

${\mu}_{1}$ | 1.92 | 0.41 | 2.00 | 0.60 |

${\mu}_{2}$ | 0.40 | 0.40 | 0.40 | 0.40 |

${\mu}_{3}$ | 1.70 | 1.70 | 0.47 | 1.15 |

${\mu}_{4}$ | – | – | 1.64 | 1.49 |

**Table A3.**Values of the optimization variables for optimized designs A and B of Figure 5.

A ($\mathit{N}=2$) | B ($\mathit{N}=2$) | A ($\mathit{N}=3$) | B ($\mathit{N}=3$) | |
---|---|---|---|---|

${k}_{0}{a}_{1}$ | 4.37 | 4.48 | 5.35 | 5.39 |

${k}_{0}{a}_{2}$ | 3.87 | 3.88 | 4.35 | 4.45 |

${k}_{0}{a}_{3}$ | – | – | 3.79 | 3.65 |

${\u03f5}_{1}$ | 3.48 | 3.03 | 0.58 | 0.65 |

${\u03f5}_{2}$ | 0.50 | 0.50 | 4.76 | 4.25 |

${\u03f5}_{3}$ | – | – | 0.80 | 0.75 |

${\mu}_{1}$ | 0.50 | 0.52 | 1.28 | 1.11 |

${\mu}_{2}$ | 1.04 | 1.04 | 0.52 | 0.56 |

${\mu}_{3}$ | – | – | 0.70 | 0.59 |

**Table A4.**Values of the optimization variables for optimized designs A and B of Figure 6.

A ($\mathit{N}=2$) | B ($\mathit{N}=2$) | A ($\mathit{N}=3$) | B ($\mathit{N}=3$) | |
---|---|---|---|---|

${k}_{0}{a}_{1}$ | 4.37 | 4.50 | 5.14 | 4.63 |

${k}_{0}{a}_{2}$ | 3.99 | 3.99 | 4.43 | 3.99 |

${k}_{0}{a}_{3}$ | – | – | 3.95 | 3.68 |

${\u03f5}_{1}$ | 3.56 | 2.90 | 0.85 | 2.43 |

${\u03f5}_{2}$ | 0.50 | 0.50 | 3.43 | 0.53 |

${\u03f5}_{3}$ | – | – | 0.50 | 0.50 |

${\mu}_{1}$ | 0.50 | 0.51 | 0.64 | 0.62 |

${\mu}_{2}$ | 1.10 | 1.16 | 1.66 | 1.39 |

${\mu}_{3}$ | – | – | 0.70 | 0.89 |

A ($\mathit{N}=2$) | B ($\mathit{N}=2$) | A ($\mathit{N}=3$) | B ($\mathit{N}=3$) | A ($\mathit{N}=4$) | B ($\mathit{N}=4$) | |
---|---|---|---|---|---|---|

${k}_{0}{a}_{1}$ | 9.68 | 9.67 | 9.69 | 10.23 | 9.82 | 9.98 |

${k}_{0}{a}_{2}$ | 7.40 | 9.00 | 9.07 | 9.35 | 9.03 | 9.20 |

${k}_{0}{a}_{3}$ | – | – | 7.52 | 7.46 | 8.19 | 9.10 |

${k}_{0}{a}_{4}$ | – | – | – | – | 6.95 | 7.60 |

${\u03f5}_{1}$ | 3.57 | 2.91 | 2.56 | 1.26 | 1.69 | 1.54 |

${\u03f5}_{2}$ | 5.00 | 5.00 | 4.42 | 4.17 | 4.73 | 4.42 |

${\u03f5}_{3}$ | – | – | 4.91 | 4.97 | 4.82 | 4.52 |

${\u03f5}_{4}$ | – | – | – | – | 4.98 | 4.39 |

${\mu}_{1}$ | 3.81 | 2.81 | 2.41 | 1.26 | 1.76 | 1.63 |

${\mu}_{2}$ | 3.76 | 3.46 | 4.31 | 4.09 | 4.78 | 4.17 |

${\mu}_{3}$ | – | – | 3.27 | 3.42 | 3.49 | 4.86 |

${\mu}_{4}$ | – | – | – | – | 3.91 | 3.11 |

**Table A6.**Values of the optimization variables for optimized designs A and B of Figure 8.

A ($\mathit{N}=2$) | B ($\mathit{N}=2$) | A ($\mathit{N}=3$) | B ($\mathit{N}=3$) | A ($\mathit{N}=4$) | B ($\mathit{N}=4$) | |
---|---|---|---|---|---|---|

${k}_{0}{a}_{1}$ | 4.89 | 4.95 | 5.27 | 5.65 | 5.67 | 5.31 |

${k}_{0}{a}_{2}$ | 3.52 | 3.51 | 4.61 | 4.82 | 5.13 | 4.97 |

${k}_{0}{a}_{3}$ | – | – | 3.69 | 4.34 | 4.75 | 4.05 |

${k}_{0}{a}_{4}$ | – | – | – | – | 3.90 | 3.58 |

${\u03f5}_{1}$ | 3.56 | 3.32 | 1.57 | 1.16 | 3.77 | 1.51 |

${\u03f5}_{2}$ | 4.60 | 4.92 | 4.57 | 2.89 | 2.66 | 4.00 |

${\u03f5}_{3}$ | – | – | 3.87 | 3.91 | 1.53 | 2.66 |

${\u03f5}_{4}$ | – | – | – | – | 4.86 | 4.98 |

${\mu}_{1}$ | 3.92 | 3.82 | 1.39 | 1.36 | 4.86 | 1.31 |

${\mu}_{2}$ | 3.67 | 3.68 | 4.78 | 4.55 | 2.92 | 3.35 |

${\mu}_{3}$ | – | – | 2.76 | 3.82 | 1.18 | 2.78 |

${\mu}_{4}$ | – | – | – | – | 3.22 | 3.68 |

**Table A7.**Values of the optimization variables for optimized designs A and B of Figure 10.

A ($\mathit{N}=2$) | B ($\mathit{N}=2$) | A ($\mathit{N}=3$) | B ($\mathit{N}=3$) | |
---|---|---|---|---|

${k}_{0}{a}_{1}$ | 5.20 | 5.20 | 5.21 | 5.26 |

${k}_{0}{a}_{2}$ | 3.47 | 3.51 | 3.74 | 4.60 |

${k}_{0}{a}_{3}$ | – | – | 3.42 | 3.47 |

${\u03f5}_{1}$ | 0.53 | 0.52 | 0.50 | 0.71 |

${\u03f5}_{2}$ | 4.90 | 4.44 | 3.93 | 0.50 |

${\u03f5}_{3}$ | – | – | 0.50 | 0.50 |

${\mu}_{1}$ | 1.68 | 1.72 | 1.90 | 1.83 |

${\mu}_{2}$ | 0.65 | 0.68 | 0.85 | 1.32 |

${\mu}_{3}$ | – | – | 0.82 | 0.67 |

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**Figure 1.**The considered scattering geometry of a layered spherical medium composed of a PEC or dielectric core of radius ${a}_{5}$ covered by four spherical shells with radii ${a}_{j}$, permittivities ${\u03f5}_{j}$, and permeabilities ${\mu}_{j}$ ($j=1,2,3,4$). The medium is excited by a magnetic dipole located at $(0,0,b)$.

**Figure 2.**Normalized bistatic SCS $\sigma (\theta ,\varphi )/\left(\pi {a}_{c}^{2}\right)$ in the $xOz$ and $yOz$ planes versus the angle $\theta $ for (

**a**,

**b**) $N=3$ and (

**c**,

**d**) $N=4$ optimized spherical layers covering a PEC core of radius ${k}_{0}{a}_{c}=2\pi $ excited by a dipole at $b=10{a}_{c}$ (nearly plane-wave incidence case).

**Figure 3.**Best fitness function’s values versus the CAPSO algorithm’s iterations for the scattering problem considered in Figure 2, presented in a stairstep graph with a 5-iteration step.

**Figure 4.**As in Figure 2 with (

**a**,

**b**) $N=3$ and (

**c**,

**d**) $N=4$, but for a dipole at $b=1.3{a}_{1}$, near the boundary of the spherical medium. The values of the optimization variables for designs A and B are given in Table A2 of the Appendix A.

**Figure 5.**Normalized bistatic SCS $\sigma (\theta ,\varphi )/\left(\pi {a}_{c}^{2}\right)$ in the $xOz$ and $yOz$ planes versus the angle $\theta $ for (

**a**,

**b**) $N=2$ and (

**c**,

**d**) $N=3$ optimized spherical shells (layers) covering a PEC core of radius ${k}_{0}{a}_{c}=\pi $ excited by a dipole at $b=10{a}_{c}$ (nearly plane-wave incidence case). The values of the optimization variables for designs A and B are given in Table A3 of the Appendix A.

**Figure 6.**As in Figure 5 with (

**a**,

**b**) $N=2$ and (

**c**,

**d**) $N=3$, but for a dipole at $b=1.3{a}_{1}$ (spherical-wave incidence case). The values of the optimization variables for designs A and B are given in Table A4 of the Appendix A.

**Figure 7.**Normalized bistatic SCS $\sigma (\theta ,\varphi )/\left(\pi {a}_{c}^{2}\right)$ in the $xOz$ and $yOz$ planes versus the angle $\theta $ for (

**a**,

**b**) $N=2$, (

**c**,

**d**) $N=3$, and (

**e**,

**f**) $N=4$ optimized spherical layers covering a dielectric core of radius ${k}_{0}{a}_{c}=2\pi $ and permittivity ${\u03f5}_{c}=2.1$, excited by a dipole at $b=10{a}_{c}$. The values of the optimization variables for designs A and B are given in Table A5 of the Appendix A.

**Figure 8.**As in Figure with (

**a**,

**b**) $N=2$, (

**c**,

**d**) $N=3$, and (

**e**,

**f**) $N=4$, but for a dielectric core with radius ${k}_{0}{a}_{c}=\pi $. The values of the optimization variables for designs A and B are given in Table A6 of the Appendix A.

**Figure 9.**Best fitness function’s values versus the CAPSO algorithm’s iterations for the problems considered in (

**a**) Figure and (

**b**) Figure 8, presented in stairstep graphs with a 5-iteration step.

**Figure 10.**Normalized bistatic SCS $\sigma (\theta ,\varphi )/\left(\pi {a}_{c}^{2}\right)$ in the $xOz$ and $yOz$ planes versus the angle $\theta $ for (

**a**,

**b**) $N=2$- and (

**c**,

**d**) $N=3$-optimized spherical layers covering a dielectric core of radius ${k}_{0}{a}_{c}=\pi $ and permittivity ${\u03f5}_{c}=18.7$, excited by a dipole at $b=10{a}_{c}$. The values of the optimization variables for designs A and B are given in Table A7 of the Appendix A.

**Figure 11.**The best fitness function’s values versus the CAPSO algorithm’s iterations for the scattering problem considered in Figure 10, presented in a stairstep graph with a 5-iteration step.

**Figure 12.**Cloaking performance for the optimized design B of Figure a,b after perturbing (

**a**,

**b**) the permittivity ${\u03f5}_{1}$ and (

**c**,

**d**) the permeability ${\mu}_{1}$ by $\pm 5\%$.

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## Share and Cite

**MDPI and ACS Style**

Michaloglou, A.; Tsitsas, N.L.
Feasible Optimal Solutions of Electromagnetic Cloaking Problems by Chaotic Accelerated Particle Swarm Optimization. *Mathematics* **2021**, *9*, 2725.
https://doi.org/10.3390/math9212725

**AMA Style**

Michaloglou A, Tsitsas NL.
Feasible Optimal Solutions of Electromagnetic Cloaking Problems by Chaotic Accelerated Particle Swarm Optimization. *Mathematics*. 2021; 9(21):2725.
https://doi.org/10.3390/math9212725

**Chicago/Turabian Style**

Michaloglou, Alkmini, and Nikolaos L. Tsitsas.
2021. "Feasible Optimal Solutions of Electromagnetic Cloaking Problems by Chaotic Accelerated Particle Swarm Optimization" *Mathematics* 9, no. 21: 2725.
https://doi.org/10.3390/math9212725